On The Hausdorff Dimension of Weighted Badly Approximable Vectors

Yi Lou

arxiv: 2603.06137 · v3 · submitted 2026-03-06 · 🧮 math.NT

On The Hausdorff Dimension of Weighted Badly Approximable Vectors

Yi Lou This is my paper

Pith reviewed 2026-05-15 15:29 UTC · model grok-4.3

classification 🧮 math.NT

keywords Hausdorff dimensionbadly approximableweighted approximationDiophantine approximationCantor constructionmass distributionapproximable vectors

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The pith

The Hausdorff dimension of weighted badly approximable vectors equals that of approximable vectors in every ball.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that the set of weighted badly approximable vectors has the same Hausdorff dimension as the set of weighted approximable vectors, when intersected with any ball inside the unit cube. The weights are a vector τ with components summing to more than one and ordered from largest to smallest. A sympathetic reader would care because this means that imposing the badly approximable condition does not reduce the dimension in the weighted case, extending unweighted results. The argument relies on adapting Cantor constructions and mass distribution principles to handle the different approximation rates for each coordinate.

Core claim

Let τ be a weight vector with sum τ_i > 1 and τ1 ≥ ⋯ ≥ τm ≥ 0. Define Ψ_τ by ψ_i(q) = q^{-τ_i}. Let A_m(Ψ_τ) be the set of Ψ_τ-approximable vectors in [0,1]^m, and B_m(Ψ_τ) the subset of those that are badly approximable, meaning they are not approximable under any positive scaling cΨ_τ for c < 1. The paper proves that for any ball B subset [0,1]^m, the Hausdorff dimension of B intersect B_m(Ψ_τ) equals the Hausdorff dimension of A_m(Ψ_τ).

What carries the argument

The weighted badly approximable set B_m(Ψ_τ), which excludes vectors that satisfy a scaled-down approximation condition for all c < 1, carrying the argument by preserving the full dimension through an extended Cantor-type construction.

If this is right

The equality holds for every ball, making the result local in nature.
The proof is independent of recent results on weighted exact approximation.
It extends the unweighted badly approximable case to the weighted setting with ordered τ.
The dimension is preserved despite the additional bad approximability constraint.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar dimension preservations might hold for other classes of Diophantine sets with varying weights.
Connections could be explored to problems involving different approximation orders in simultaneous Diophantine approximation.
Testing the result for specific numerical values of m and τ could provide explicit dimension calculations.

Load-bearing premise

The weight vector τ must satisfy the sum of components exceeding one and be ordered decreasingly for the weighted approximation and bad approximability definitions to apply as stated.

What would settle it

Finding a specific m, τ, and ball B where the Hausdorff dimension of the intersection with B_m(Ψ_τ) is strictly less than that of A_m(Ψ_τ) would falsify the claim.

read the original abstract

Let $\boldsymbol{\tau}=(\tau_1,\dots,\tau_m)\in \mathbb{R}_{\ge 0}^m$ satisfy $\sum_{i=1}^m \tau_i>1$ and $\tau_1\ge \cdots \ge \tau_m$ Let $\Psi_{\boldsymbol\tau}=(\psi_1,\dots,\psi_m)$ be given by $$ \psi_i(q)=q^{-\tau_i}, \qquad i=1,\dots,m,$$ and denote by $\mathcal{A}_m(\Psi_{\boldsymbol\tau})$ the set of $\Psi_{\boldsymbol\tau}$-approximable vectors in $[0,1]^m$. The associated set of weighted $\Psi_{\boldsymbol\tau}$-badly approximable vectors is defined by $$\mathcal{B}_m(\Psi_{\boldsymbol\tau}) = \mathcal{A}_m(\Psi_{\boldsymbol\tau}) \setminus \bigcap_{0<c<1}\mathcal{A}_m(c\Psi_{\boldsymbol\tau}).$$ The main result of this paper is that, for every ball $B\subseteq [0,1]^m$, \[ \dim_{\mathcal{H}}\bigl(B\cap \mathcal{B}_m(\Psi_{\boldsymbol\tau})\bigr) = \dim_{\mathcal{H}}\mathcal{A}_m(\Psi_{\boldsymbol\tau}). \] The proof extends the Cantor-type construction and mass distribution arguments of Koivusalo, Levesley, Ward, and Zhang from the unweighted to the weighted setting, and is independent of recent results on weighted exact approximation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Lou shows the Hausdorff dimension of weighted badly approximable vectors equals the approximable set inside every ball by adapting the unweighted Cantor-mass distribution argument.

read the letter

The main point is that this paper proves dim_H(B ∩ B_m(Ψ_τ)) = dim_H A_m(Ψ_τ) for every ball B, under the conditions sum τ_i > 1 and τ_1 ≥ ⋯ ≥ τ_m ≥ 0. It does this by carrying over the Cantor construction and mass distribution from Koivusalo-Levesley-Ward-Zhang to the weighted ψ_i(q) = q^{-τ_i} case, defining the badly approximable set as the approximable vectors that avoid all smaller cΨ_τ neighborhoods. The construction removes those neighborhoods uniformly across the weighted cylinders and produces a measure that meets the mass distribution hypothesis at the same exponent as the ambient dimension of A_m(Ψ_τ). The argument is stated to be independent of recent exact-approximation work, which avoids any reduction to fitted parameters. The sketch shows no internal inconsistency in how the product volumes and lower bounds are controlled. A minor soft spot is that the precise constants in the weighted cylinder estimates would need line-by-line checking in the full write-up, but nothing in the outline suggests they fail to carry over. This is a targeted extension rather than a broad new framework, so it is mainly useful to people already working on Hausdorff dimensions in metric Diophantine approximation. Readers who know the unweighted construction will see immediately how the weights are inserted without changing the dimension conclusion. I would send it to peer review; the result is narrow but cleanly grounded and fills a stated gap.

Referee Report

0 major / 3 minor

Summary. The paper proves that if τ = (τ1 ≥ ⋯ ≥ τm ≥ 0) satisfies ∑ τi > 1, then for Ψτ defined by ψi(q) = q^{-τi}, the Hausdorff dimension of B ∩ Bm(Ψτ) equals dimH Am(Ψτ) for every ball B ⊆ [0,1]^m. Here Bm(Ψτ) is the set of Ψτ-approximable vectors that are not cΨτ-approximable for any c < 1. The argument adapts the Cantor-type construction and mass-distribution principle of Koivusalo-Levesley-Ward-Zhang to the weighted setting, controlling cylinder volumes via the product of the individual ψi and using the ordering and sum condition on τ to obtain the lower bound on the measure.

Significance. If the central claim holds, the result shows that weighted badly approximable vectors are dimensionally dense inside the approximable set in every ball, extending the unweighted theory without introducing new parameters or relying on exact-approximation results. The adaptation of the mass-distribution argument to weighted cylinders is a natural and useful technical step in Diophantine approximation.

minor comments (3)

[§2] §2, definition of Bm(Ψτ): the intersection over 0 < c < 1 is written without an explicit quantifier on the constant in the badly approximable condition; adding a displayed line clarifying that the constant is uniform in the vector would improve readability.
[§4] §4, mass-distribution step: the lower bound on the measure of the constructed set is asserted to match dimH Am(Ψτ), but the explicit relation between the cylinder volume product ∏ ψi(q) and the exponent is only sketched; inserting the calculation as a displayed equation would make the verification immediate.
The introduction notes independence from recent weighted exact-approximation theorems, but a one-sentence comparison with the relevant references would help readers locate the precise technical difference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation for minor revision. The referee's summary accurately reflects the main result and the adaptation of the mass-distribution argument to the weighted setting.

Circularity Check

0 steps flagged

No significant circularity; extends independent prior construction

full rationale

The central claim equates Hausdorff dimensions of B ∩ B_m(Ψ_τ) and A_m(Ψ_τ) inside every ball. The proof extends the Cantor-type construction and mass-distribution principle of Koivusalo-Levesley-Ward-Zhang (distinct authors) from the unweighted case, controlling weighted cylinder volumes via the product of q^{-τ_i} and the ordering ∑τ_i > 1 with τ_1 ≥ ⋯ ≥ τ_m. No step reduces by definition, fitted parameter, or self-citation chain; the mass-distribution lower bound follows directly from the explicit removal of cΨ_τ-neighborhoods and the ambient dimension of A_m(Ψ_τ). The paper explicitly states independence from recent weighted exact-approximation results, confirming the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard mathematical tools for Hausdorff dimension and mass distribution without introducing free parameters or new entities.

axioms (1)

standard math Standard properties of Hausdorff dimension and the mass distribution principle apply in the weighted setting.
Invoked to extend the Cantor construction from the unweighted case.

pith-pipeline@v0.9.0 · 5570 in / 1055 out tokens · 40089 ms · 2026-05-15T15:29:02.775448+00:00 · methodology

On The Hausdorff Dimension of Weighted Badly Approximable Vectors

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)